low cards in the flop (O8/B )
 

Anybody ever calculate the probability of flopping 0,1,2 or 3 low cards (A-8)? I'm reading Cappelletti's O/8B book and he says that those probabilities, when you hold A24Q, are 5.6%, 33.1%, 46% and 15.3%. I agree with the first number (19 choose 3 = 5.6%) but can't figure on the last three. Buzz, Felicia or anybody else out there, can you help?

Re: low cards in the flop (O8/B )
 

[ QUOTE ]
Anybody ever calculate the probability of flopping 0,1,2 or 3 low cards (A-8)? I'm reading Cappelletti's O/8B book and he says that those probabilities, when you hold A24Q, are 5.6%, 33.1%, 46% and 15.3%.

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Domester - We don't exactly want the probability of flopping 0, 1, 2, or 3 low <font color="white">_</font>cards. Of more interest is the probability of flopping 0, 1, 2, or 3 low <font color="white">_</font>ranks

The answer to the number of low cards is very easy:
• 0 low cards on flop: P = 0.0560
• 1 low card on flop: P = 0.2867
• 2 low cards on flop: P = 0.4460
• 3 low cards on flop: P = 0.2113

But this is misleading because when there are two low cards on the flop, the two low cards could be a pair, together amounting to only one rank of low card.
Similarly when there are three low cards on the flop we could have a pair or trips.

The answer to the number of low ranks is more tedious:
• 0 different ranked low cards on flop: P = 969/17296 = 0.0560
• 1 rank of low cards on flop: P = 5723/17296 = 0.3309
• 2 different ranks of low cards on flop: P = 7957/17296 = 0.4600
• 3 different ranks of low cards on flop: P = 2647/17296 = 0.1530

Those are evidently the numbers Cappelletti is using.

But, since you’ve brought it up, aren’t we really interested in knowing when we have two or three low cards on the flop that are of benefit to us?

For example, there are some flops with three different ranks of low cards that don’t make a low for us when we hold A24Q. (eg. 2-4-8).
And there are other flops three different ranks of low cards that make a low, but not the nut low. (eg. 2-7-8).

Does he later somehow make a correction for this?

Buzz

Re: low cards in the flop (O8/B )
 

[ QUOTE ]
Anybody ever calculate the probability of flopping 0,1,2 or 3 low cards (A-8)?

[/ QUOTE ]

Cappelletti says "What percent of the time do three cards contain zero, one, two, or three different low cards?" "Different" is the key word here. Use that and you should get his numbers.

Sam

Re: low cards in the flop (O8/B )
 

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But, since you’ve brought it up, aren’t we really interested in knowing when we have two or three low cards on the flop that are of benefit to us?

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He talks about a similar situation is the beginning of Chapter 4, although I'm still trying to reconcile all the numbers he throws around.

Sam

Re: low cards in the flop (O8/B )
 

Buzz,

He doesn't make a later correction for cards that can help. It just says three low cards. If you can please provide calculations for the numerator of those last three percentages, I'd much appreciate it. I still can't make my numbers work. Thanks.

Re: low cards in the flop (O8/B )
 

[ QUOTE ]
But, since you’ve brought it up, aren’t we really interested in knowing when we have two or three low cards on the flop that are of benefit to us?

[/ QUOTE ]

Buzz, can you check my numbers?

Holding AQ42 as an example of any low-low-low-high hand:

The flop contains exactly two low cards that are not A, 2, or 4 AND are not themselves paired 17.6% of the time.

The flop contains exactly two low cards, one of which IS an A, 2, or 4, and one of which is not A, 2, or 4 9.9% of the time.

Sam

Re: low cards in the flop (O8/B )
 

Sam,

I guess my issue is the uncertain meaning of "different". Is it different from A24, or simply different ranks from each other? For example, for one low card, my math is as follows... there are 29 of them left in the deck and there are (29 choose 2 =) 171 ways to pick two high cards, so 29*171/17,296 = 28.7% chance of flopping one low card. If you exclude the As, 2s and 4s, the calc is 20*171/17,296 = 19.8% chance of flopping one low card. Neither one is close to Cappelletti's answer. Am I smoking the wacky weed on this one?

Re: low cards in the flop (O8/B )
 

For one low card, there are 29 left in the deck. But once you pick that low card (let’s say it’s the 3c), the other low cards of that rank (3s,3d,3h) become high cards for purposes of this calculation. So odds of 1 low card would be C(29,1) * C(22,2). But that doesn’t take into account the slight adjustment you have to make for picking a low that is in your hand vs. not in your hand. And once you make that calculation, you need to adjust out since picking 3s,3c, T is the same as 3c, 3s, T. Alternatively, calculate C(29,1) * C(19,2) then add in times you choose the same ranked low card with the 2nd or 3rd selection.

Re: low cards in the flop (O8/B )
 

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If you can please provide calculations for the numerator of those last three percentages, I'd much appreciate it. I still can't make my numbers work. Thanks.

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Domester - Numerator for the last three percentages? Sure. I didn’t save my scratchwork after I computed and posted. Not sure I’ll do it the same way I did last time. But in any event, what you do is make a chart showing the various possibilities.

You hold A24Q. The low cards in the stub are
3 aces, 3 deuces, 3 fours,
4 treys, 4 fives, 4 sixes, 4 sevens and 4 eights.
The other 19 cards in the stub are all high cards.

After taking out (mentally) of the deck your four card hand there are three cards left in the stub for each of the three low ranks in your hand, and four cards left in the stub for each of the five low ranks not in your hand. There are also nineteen high cards left in the stub.
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• 1 rank of low cards on flop. In this case, there must be either one card, two cards, or three cards of a single low rank. The other flop cards must be high cards.
For a single ace, deuce, or four plus two high cards: 3*19*18/2
For a pair of aces, deuces, or fours plus one high card: 3*19
For trip aces, deuces, or fours: 1

For a single trey, five, six, seven or eight plus two high cards: 4*19*18/2
For a pair of treys, fives, sixes, sevens or eights plus one high card: 6*19
For trip treys, fives, sixes, sevens or eights: 4*1

Computing:
3*(513+57+1) = 1713 ways to have a flop with a single rank of aces, deuces, or treys.
5(684+114+4) = 4010 way to have a flop with a single rank of treys, fives, sixes, sevens or eights.
Then 1713+4010 = 5723.

What I just did seems totally strange to me. I must have thought of the solution differently when I answered your prefious post. But in any event, the numerator I’m getting is the same.

Instead of my spending the rest of the night trying to figure out how to make the above response clearer to you, ask about what you don’t understand and maybe I (or somebody else) can explain it to you. Basically, I’m listing the number of possible different combinations that satisfy the requirement of exactly one low rank on the flop, and I’m doing that for each of the different low ranks.
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• 2 ranks of low cards on flop. In this case, there must be either a single card of each of two low ranks plus a high card, or there must be a pair of one low rank and a single card of another low rank.

There are two groups of low ranks.

*first group: 3 aces, 3 deuces, 3 fours,
*second group: 4 treys, 4 fives, 4 sixes, 4 sevens and 4 eights.
*The other 19 cards in the stub are all high cards.

Here are all the possibilities:
(d.r. means different rank)
(s.c. means single card)
first group pair + d.r. first group s.c.,
first group pair + second group s.c.,
second group pair + first group s.c.,
second group pair + d.r. second group s.c.,
high card + first group s.c. + d.r. first group s.c.
high card + first group s.c. + second group s.c.
high card + second group s.c. + d.r. second group s.c.

That’s a chart, of sorts, a seven line chart. Honestly it looks contorted to me. last time I did it, as I recall, it was neater. Oh well.
Now I’ll write the number of possibilities for each line. Here goes:
3*3*6 = 54
3*3*20 = 180
5*6*9 = 270
5*6*16 = 480
19*9*6/2 = 513
19*9*20 = 3420
19*20*16/2 = 3040

Then 54+180+270+480+513+3420+3040 = 7957.

That sure looks contorted to me, but lo and behold, it’s the same answer I got before. You can see it’s a bitch doing these. Lots of places to go wrong.
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• 3 ranks of low cards on flop.
O.K. here comes the last one. This is actually going to be pretty easy. All three different ranks from group one or group two, then two ranks from one group and one rank from the other.
This one’s sort of a four line chart.
I’ll just write the numbers that lead to an answer without an explanation. Ask if you don’t understand and I’ll try to explain.

20*16*12/6
9*6*3/6
9*20*16/2
20*9*6/2
Computing, 640+27+1440+540 = 2647.

Anyone wants to explain more clearly, be my guest.

Buzz

Re: low cards in the flop (O8/B )
 

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Buzz, can you check my numbers?

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Sure, Sam.

[ QUOTE ]
Holding AQ42 as an example of any low-low-low-high hand:

The flop contains exactly two low cards that are not A, 2, or 4 AND are not themselves paired 17.6% of the time.

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(19*20*16/2)/(48*47*46/6) =
3040/17296 = 0.1758

Looks like we agree on the first one.

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The flop contains exactly two low cards, one of which IS an A, 2, or 4, and one of which is not A, 2, or 4 9.9% of the time.

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But I get a different answer (19.77%) for the second one. (as shown below):

9*20*19/17296 = 3420/17296 = 0.1977

Buzz